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International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962
Volume-12, Issue-1, (February 2022)
www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15
115 This work is licensed under Creative Commons Attribution 4.0 International License.
Modal Space Controller for Hydraulically Driven Six Degree of
Freedom Parallel Manipulator
P. O. Ogbobe1
and C. N. Okoye2
1
Department of Information and Communication Technology, National Board for Technology Incubation, Abuja, NIGERIA
2
Federal Ministry of Education, Abuja, NIGERIA
1
Corresponding Author: peter.ogbobe@nbti.gov.ng
ABSTRACT
This paper presents the Modal space decoupled
control for a hydraulically driven parallel mechanism has
been presented. The approach is based on singular values
decomposition to the properties of joint-space inverse mass
matrix, and mapping of the control and feedback variables
from the joint space to the decoupling modal space. The
method transformed highly coupled six-input six-output
dynamics into six independent single-input single-output
(SISO) 1 DOF hydraulically driven mechanical systems. The
novelty in this method is that the signals including control
errors, control outputs and pressure feedbacks are
transformed into decoupled modal space and also the
proportional gains and dynamic pressure feedback are tuned
in modal space. The results indicate that the conventional
controller can only attenuate the resonance peaks of the lower
eigenfrequencies of six rigid modes properly, and the peaking
points of other relative higher eigenfrequencies are over
damped, The further results show that it is very effective to
design and tune the system in modal space and that the
bandwidth increased substantially except surge (x) and sway
(y) motions, each degree of freedom can be almost tuned
independently and their bandwidths can be increased near to
the undamped eigenfrequencies.
Keywords-- Hydraulically Driven, 6 DOF Parallel
Mechanism, Modal Space, Dynamic Pressure Feedback
I. INTRODUCTION
Modal space control has in the recent time
become popular as a control strategy in linear structural
dynamics and related fields like aero elasticity and active
vibration control and robotics. This type of control concept
originated from process control problems, which were
summarized in Porter and Crossley [1]. Gould and
Muarray-Lasso intended the modal control concepts to
structure systems [2]. Meirovitch [3] developed an
independent modal space control (IMSC) method for
controlling a single mode of a distributed mass body. In
IMSC, the equations of motion for the structure are
decoupled using modal analysis and then modal control
forces are determined by minimizing a performance index.
Baz and Poh [4] in their study performed a numerical
study on controlling the vibrations of a beam instrumented
with piezoelectric patches, using IMSC. They modified the
IMSC to control multiple modes and called the result
modified independent modal space control (MIMSC). Baz
and Poh [5] experimentally verified the theoretical
developments by controlling the first two modes of a
cantilevered beam.
Other notable works can be found in Anthonis [6].
He applied modal space decoupling control to control the
active suspension of a spray boom. The two hydraulic
actuators of the active suspension are placed symmetrically
on the spray boom. Because the spray boom itself is also
symmetrical, the two excitations can be transformed to a
translational and rotational excitation. This modal
transformation yields perfect decoupling in the frequency
range where only the rigid body modes determine the
vibration. Lauwerys, Swevers and Sas [6] used a similar
approach to control an active suspension of a car. The
motion of the car body is transformed in heave, roll and
pitch motions which are decoupled. Lin and Yu [7] used
modal decoupling to suppress the vibration of a rotor.
The popularity of modal space control is due to its
numerous benefits. One of the most important advantages
is its ability to reduce the order of a system by choosing
the most important modes of the system. Also, the natural
modes decouple the structural system and further simplify
the system analysis, no optimization is required and
numerically stable methods exist to determine the modal
vectors. In addition, a tremendous insight is gained into the
characteristics and behavior of the system [8].
While significant progress has been made in the
field of modal space control concept, the application of
modal space to the control of hydraulically driven 6 DOF
motion simulation platform has not been fully exploited. In
this study the application of modal space control strategy,
which is receiving intense interest in the control of multi-
degree-of-freedom (MDOF) systems and has been
successively applied to a wide variety of practical
problems, is examined. It is the purpose of this work to
extend the modal space control to 6 DOF motion
simulation platform by using modal space decoupling
control strategy (MSDC). The methodology is based on
exploitation of the properties of the joint space inverse
International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962
Volume-12, Issue-1, (February 2022)
www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15
116 This work is licensed under Creative Commons Attribution 4.0 International License.
mass matrix. Through a mapping of the control and
feedback variables, from the joint space to the decoupling
modal space, the highly coupled MDOF dynamics is
transformed into six independent single-input single-output
(SISO) 1 DOF hydraulically driven mechanical system.
II. SIX DEGREES OF FREEDOM
PARALLEL MANIPULATOR (PM)
DYNAMICS
The 6 DOF PM is a parallel mechanism that
consists of a rigid body top plate, or mobile plate,
connected to a fixed base plate and is defined by at least
three stationary points on the grounded base connected to
six independent kinematic legs. The six legs of the 6 DOF
PM are connected to both the base plate and the top plate
by universal joints in parallel located at both ends of each
leg. The legs are designed with an upper body and lower
body that can be adjusted, allowing each leg to be varied in
length. Each leg subsystem contains two bodies connected
together with a cylindrical joint. The position and
orientation of the mobile platform varies depending on the
lengths to which the six legs are adjusted. The overall
system has six degrees of freedom. The mechanism is
depicted pictorially in figure 1.
Lower gimbal
point
Moving platform (p)
Servovalve
Fixed base (b)
Actuator
Hydraulic cylinder
Upper gimbal
point
Rod
Figure 1: Hydraulically Driven 6 DOF Parallel Manipulator
III. SIX DEGREES OF FREEDOM PM
KINEMATICS
The kinematics analysis that establishes the
relationship between the lengths of the six actuators and
the position of the moving platform will be presented.
There are two main axis systems used to describe the
motion: body-fixed and earth-fixed coordinates. The
calculations are done in the body-fixed axis to simplify the
equations; the results are then transformed to the earth-
fixed coordinate system for proper updating of position
and orientation. The earth-fixed system is considered an
inertial reference frame, neglecting the rotational velocity
of the earth and conforming to Newton’s laws of motion. A
flat earth assumption is made, which essentially aligns the
acceleration of gravity along the vertical earth-fixed
system.
In figure 3 a schematic drawing of the coordinate
system is shown, addressing the geometric relations of the
system with coordinates of 6 DOF PM. The six joints are
on the center Ai (i = 1, 2... 6), located in the circle with
radius ra, and denoted by A1, A3, A5 and A2, A4, A6.
The center of the six joints is connected to form a
hexagon. Similarly, six of the attachment points Bi (i = 1,
2..., 6), located in the circle of radius rb, and B1, B3, B2
and B4, B5, B6, respectively constitute two equilateral
triangles. The connection of the upper and lower
attachment points form a platform rotated 180 degrees
with respect to each other. The motion of the moving
platform is generated by actuating the cylindrical joints
which vary the lengths of the legs, li, i = 1….6.
International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962
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117 This work is licensed under Creative Commons Attribution 4.0 International License.
6
A
1
B
5
A
4
A
3
A
2
l
a
d
p
O
2
A
2
B
3
B
4
B
5
B
6
B
b
d
b
r
a
r
)
( b
p X
X
)
( b
p Y
Y
p
Y
p
X
b
X
b
Y
b
Z
p
Z
1
A
2
A
3
B
2
B
4
B
5
B
1
B
6
B
b
O
2
l
1
2
3
4
5 6
1
2
3
4
5
6
p
O
3
A
4
A 5
A
6
A
1
A
1
α
2
α
a) Front view b) top view
Figure 3: Schematic diagram of coordinate system of 6-DOF PM
So, trajectory of the center point of moving
platform is adjusted by using these variables. For modeling
of the mechanism, a base reference frame {B} (OB-XB-YB-
ZB) is defined as shown in figure 3. A second frame {p}
(Op-Xp-Yp-Zp) is attached to the center of the moving
platform, Op and the points linking the legs to the moving
platform are noted as Ai, i = 1….6, and each leg is attached
to the base platform at the point Bi, i = 1….6. At neutral
pose the body axes {p} attached to the moving platform
are parallel to and coincide with the inertial frame {B}
fixed to the base with its origin at the geometric centre of
the base platform. Kinematics and dynamics analysis of
the 6 DOF parallel mechanisms has been well established
and can be found in several literatures [9-13].
3.1 Control Design
The structure of the proposed modal space
controller is similar to the conventional joint space control
as shown in figure 4. However, the difference is that the
signals including control errors, control outputs and
pressure difference feedbacks are transformed into
decoupled modal space, so that the coupling effects are
fully considered and compensated. Thus, the proportional
gains and dynamic pressure feedback functions can be
tuned independently in modal space. Also, the bandwidth
can be raised near to the undamped eigenfrequencies.
Figure 4: Structure of the modal space decoupled controller
 
T
des z
y
x
x 

 ,
,
,
,
,

i
p
i
i b
t
Ra
l 


Inverse
Kinematics
Servo
Valve
p
K
L
P
s
1
dp
K
l
c

1
+
-
+ -
+
-
T
U
T
U
U
Dynamic pressure
feedback correction
Actuator position
Pressure feedback
Desired Input reference
Transformation
matrix
G
J
A
K
c T
lq
2
1
q
tc 1

+
+
Gravity force
compensation
International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962
Volume-12, Issue-1, (February 2022)
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118 This work is licensed under Creative Commons Attribution 4.0 International License.
Based on this model, the control inputs of the
servo valves can be developed. It can be expressed as:
T T
a,1 a,2 a,6
T
,1 ,2 ,6 T
,1 ,2 ,6
,1 ,2 ,6
T 1
2
1
diag( )
s s s
diag
s 1 s 1 s 1
i
c c c
dp dp dp
c c c
tc
lq
q
k k k
k k k
c
K A

 
  
 
 
  
 
  
     
 
 
 
 

L
i U U e
U U P
J G
(1)
com
 
e l l (2)
Where e —— position error (m),
com
l i —— actuator length command in terms of inverse kinematics (m).
And each element of i
i can be give as
6 6 6
,k
L 2
1 1 1 , 1
s
s 1 i
c tc
ik jk a,k i ik jk dp,k ij i
i k k c k q
c
U U k U U k
K A


  
 
 
 
  
 
 
   
 

   
 
  
i
i e P J G (3)
Where e ——
i
e (m),
i
U ——
 ,
jk
U
L
P ——
Li
P (Pa),
1
T -
lq
J ——
 ,
ij
J
G —— ( )
i
G (m/s2
).
When 6
,
2
,
1
, ... a
a
a k
k
k 
 and,
6
,
2
,
1
, ... dp
dp
dp k
k
k 
 and unitary orthogonal
matrix, U equals one, then, the modal space controller
degenerates into the conventional joint space controller.
IV. RESULTS AND DISCUSSION
In this section, a particular test case representing
typical physical conditions are designed to demonstrate the
performance and feasibility of the decoupling control
strategy by integrating and testing our methodologies, and
numerical simulation results from the PM is realized in the
Matlab/Simulink environment. A comparison was being
made between the modal space controller and the
conventional controller to discuss the effectiveness or
otherwise of both controllers.
The evaluation of control performance of the 6
DOF PM as applicable to flight motion simulators, most
often deals with the describing function. The describing
function can be in the form of measuring the frequency
response such that the magnitude of the response is flat.
These description functions enable the characterization of
some of the parameters which are considered most
important in control such as bandwidth, damping and
interaction effects in multivariable system [14].
The results of Bode diagrams of the closed loop
system from the desired to actual positions applying the
modal space controller are shown in figures 5 and 6
respectively.
International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962
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119 This work is licensed under Creative Commons Attribution 4.0 International License.
Figure 5: Simulation result of modal space controlled motion system frequency response
Figure 6: Simulation result of modal space controlled motion system frequency response
There are several characteristics of the system
that can be read directly from this Bode plot. For example,
in figures 5 and 6, the -3dB point can be found around
71.2Hz for pitch (Ry) and roll (Rx), 66.7Hz for heave (z),
29.4Hz for yaw. In surge(x) and sway (y) the bandwidth is
just only 18.1Hz which is the lowest compared to other
directions. The -90 deg bandwidth can be found at 28.7Hz
for pitch (Ry) and roll (Rx), 27.7Hz for heave (z), 17.2Hz
for yaw (Rz), 11.2Hz for surge(x) and sway(y). Highest
bandwidths are attained with pitch (Ry), roll (Rx) and
lowest in surge(x) and sway(y). All responses are
reasonably flat without peaking above +3dB. As
mentioned earlier, our analysis and comparing is based on
the performance of the controllers on hydraulically driven
10
1
10
2
-180
-90
0
Phase
(deg)
Frequency (Hz)
-40
-30
-20
-10
0
10
Magnitude
(dB)
Ψ
z
φ,θ
z
x,y
φ,θ
x,y Ψ
10
1
10
2
-180
-90
0
Phase
(deg)
Frequency (Hz)
-40
-30
-20
-10
0
10
Magnitude
(dB)
Ψ
z
φ,θ
z
x,y
φ,θ
x,y Ψ
10
1
10
2
-180
-90
0
Phase
(deg)
Frequency (Hz)
-40
-30
Ma
z
φ,θ
x,y Ψ
10
1
10
2
-180
-90
0
Phase
(deg)
Frequency (Hz)
-40
-30
Ma
z
φ,θ
x,y Ψ
International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962
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120 This work is licensed under Creative Commons Attribution 4.0 International License.
6 DOF PM as it relates to flight motion simulator. The
results obtained from modal space control strategy are
compared with conventional joint space controller for the
dynamic responses of the closed loop system. The
frequency response for the conventional joint space
controller is given in Figures 7 and 8.
Figure 7: Simulation result of conventional PID controlled motion system frequency response
Figure 8: Simulation result of conventional PID controlled motion system frequency response
Because there is no compensation of coupling
effects for the different natural frequencies, the responses
of surge (x) and sway (y), demonstrate peaks much more
than others (such as pitch (Ry) and roll (Rx), which are
over damped). Except for surge and pitch, the bandwidth is
as low as 8.4Hz.
Moreover, a compromise has to be found between
peaking (up till 3dB) of the lowest bandwidth loops of
surge (x) and sway (y) and the over damped response of
pitch (Ry) and roll (Rx). For this control structure, the
control action is being limited by the lowest
eigenfrequencies of the system, therefore, the maximum
attainable bandwidth is 8.5Hz for both roll (Rx) and pitch
(Ry). With the conventional controller the attainable
bandwidth for x and y is 21.8Hz, which is higher than with
modal space controller which is only 18.1Hz.
-40
-30
-20
-10
0
10
20
Magnitude
(dB)
10
1
10
2
-270
-180
-90
0
Phase
(deg)
Frequency (Hz)
x,y
z
φ,θ
Ψ
x,y
Ψ
z
φ,θ
-40
-30
-20
-10
0
10
20
Magnitude
(dB)
10
1
10
2
-270
-180
-90
0
Phase
(deg)
Frequency (Hz)
x,y
z
φ,θ
Ψ
x,y
Ψ
z
φ,θ
-40
-30
-20
-10
0
Magnitude
(dB)
10
1
10
2
-270
-180
-90
0
Phase
(deg)
Frequency (Hz)
x,y φ,θ
Ψ
x,y
Ψ
z
φ,θ
International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962
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121 This work is licensed under Creative Commons Attribution 4.0 International License.
Nevertheless, this does not suggest that the dynamic
performance along these degrees of freedom is deteriorated
using modal space controller.
As can be observed in figure 9, the conventional
control bandwidth is achieved at the expense of higher
overshoot and poor performances of other degrees of
freedom, however, a high percentage of overshot is an
undesirable phenomenon in flight motion simulator.
Moreover, the bandwidth of the modal space controller can
be tuned to a higher level, as the same as the conventional
control, but that will not meet the requirements of a flight
simulator motion system in which flat frequency response
is preferable [15]. In addition, with the modal space
controller, obviously each degree of freedom can be almost
tuned independently, and it is possible to raise their
bandwidths near to the undamped eigenfrequencies.
Comparing figure 5 and figure 6 we can see that the
bandwidth of roll and pitch increased from 8.5Hz to
71.1Hz, yaw is raised from 8.4Hz to 29.4Hz, and heave
increased from 9.2Hz to 66.7Hz. The comparison of two
control strategies indicates that the modal space control
performed better than the conventional.
Transients’ response results as shown in figure 9
and 10 are also provided to show characteristics of the
system in time domain. Step inputs are fed into the system
along each DOF in proper sequence. Inputs amplitude is 1
deg along rotation axes and 3 mm along translation axes
respectively.
a) roll
d) roll
b) pitch e) pitch
1 1.5 2 2.5
-0.02
-0.01
0
0.01
0.02
Time (s)
Amplitude
(rad)
1 1.5 2 2.5
-0.02
-0.01
0
0.01
0.02
Time (s)
Amplitude
(rad)
Ref



1 1.5 2 2.5
-0.02
-0.01
0
0.01
0.02
Time (s)
Amplitude
(rad)
Ref



1 1.5 2 2.5
-0.02
-0.01
0
0.01
0.02
Time (s)
Amplitude
(rad)
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122 This work is licensed under Creative Commons Attribution 4.0 International License.
c) yaw
f) yaw
Figure 9: Simulation result of short time response with step position inputs of the modal space control (left column, a-c)
and conventional control (right column, d-f)
a) surge
d) surge
b) sway e) sway
1 1.5 2 2.5
-0.02
-0.01
0
0.01
0.02
Time (s)
Amplitude
(rad)
1 1.5 2 2.5
-0.02
-0.01
0
0.01
0.02
Time (s)
Amplitude
(rad)
3 3.5 4 4.5
-4
-2
0
2
4
x 10
-3
Times (s)
Amplitude
(m)
1 1.5 2 2.5
-5
0
5
x 10
-3
Time (s)
Amplitude
(m)
3 3.5 4 4.5
-4
-2
0
2
4
x 10
-3
Times (s)
Amplitude
(m)
1 1.5 2 2.5
-5
0
5
x 10
-3
Time (s)
Amplitude
(m)
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3.5 4 5
-4
-2
0
4
Time (s)
Amplitude
(m)
2
x 10
-3
-3 4.5
c) heave
f) heave
Figure 10: Simulation result of short time response with step position inputs of the modal space control (left column, a-c)
and conventional control (right column, d-f)
V. CONCLUSION
Modal space decoupled control for a hydraulically
driven motion simulation platform has been presented. The
approach is based on singular values decomposition to the
properties of joint-space inverse mass matrix, and mapping
of the control and feedback variables from the joint space
to the decoupling modal space. The method transformed
highly coupled six-input six-output dynamics into six
independent single-input single-output (SISO) 1 DOF
hydraulically driven mechanical systems. It has also been
shown that the methods well suited for 1 DOF
hydraulically driven mechanical systems can be applied to
control a multi input multi output and highly coupled
hydraulically driven 6 DOF motion simulation platform.
Although, the structure of our proposed modal
space controller is very similar to the conventional joint
space control, the novelty in our approach is that the
signals including control errors, control outputs and
pressure feedbacks are transformed into decoupled modal
space and also the proportional gains and dynamic
pressure feedback are tuned in modal space. The results
indicate that the conventional controller can only attenuate
the resonance peaks of the lower eigenfrequencies of six
rigid modes properly, and the peaking points of other
relative higher eigenfrequencies are over damped, so the
system bandwidth is limited by the lower
eigenfrequencies. In addition, the results show that it is
very effective to design and tune the system in modal
space. The bandwidth increased substantially except surge
(x) and sway (y) motions, each degree of freedom can be
almost tuned independently and their bandwidths can be
increased near to the undamped eigenfrequencies.
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1 1.5 2 2.5
-4
-2
0
2
4
x 10
-3
Time (s)
Amplitude
(m)
International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962
Volume-12, Issue-1, (February 2022)
www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15
124 This work is licensed under Creative Commons Attribution 4.0 International License.
[11] P. Ogbobe, Y. Zhengmao, H, Jiang, C. Yang, & J.
Han. (2011). Formulation and evaluation of coupling effect
between DOF motions of hydraulically driven 6 DOF
parallel manipulator. Iranian Journal of Science &
Technology, Transaction of Mechanical Engineering,
35(2), 143-175. DOI: 10.22099/ijstm.2011.902.
[12] Zhang, C. & Jiang, H. (2021). Rigid-flexible modal
analysis of the hydraulic 6-dof parallel mechanism.
Energies, 14, 1604. https://doi. org/10.3390/en14061604.
[13] Gosselin CM. (1998). A new approach for the
dynamic analysis of parallel manipulator. Multibody
System Dynamics, 2, 317.
[14] Meirovitch, L. (1986). Elements of vibration analysis.
(2nd
ed.). New York, USA: McGraw-Hill.
[15] Koekebakker, S.H. (2001). Model based control of a
flight simulator motion system. PhD Thesis, Delft
University Technology.

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Modal Space Controller for Hydraulically Driven Six Degree of Freedom Parallel Manipulator

  • 1. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 115 This work is licensed under Creative Commons Attribution 4.0 International License. Modal Space Controller for Hydraulically Driven Six Degree of Freedom Parallel Manipulator P. O. Ogbobe1 and C. N. Okoye2 1 Department of Information and Communication Technology, National Board for Technology Incubation, Abuja, NIGERIA 2 Federal Ministry of Education, Abuja, NIGERIA 1 Corresponding Author: peter.ogbobe@nbti.gov.ng ABSTRACT This paper presents the Modal space decoupled control for a hydraulically driven parallel mechanism has been presented. The approach is based on singular values decomposition to the properties of joint-space inverse mass matrix, and mapping of the control and feedback variables from the joint space to the decoupling modal space. The method transformed highly coupled six-input six-output dynamics into six independent single-input single-output (SISO) 1 DOF hydraulically driven mechanical systems. The novelty in this method is that the signals including control errors, control outputs and pressure feedbacks are transformed into decoupled modal space and also the proportional gains and dynamic pressure feedback are tuned in modal space. The results indicate that the conventional controller can only attenuate the resonance peaks of the lower eigenfrequencies of six rigid modes properly, and the peaking points of other relative higher eigenfrequencies are over damped, The further results show that it is very effective to design and tune the system in modal space and that the bandwidth increased substantially except surge (x) and sway (y) motions, each degree of freedom can be almost tuned independently and their bandwidths can be increased near to the undamped eigenfrequencies. Keywords-- Hydraulically Driven, 6 DOF Parallel Mechanism, Modal Space, Dynamic Pressure Feedback I. INTRODUCTION Modal space control has in the recent time become popular as a control strategy in linear structural dynamics and related fields like aero elasticity and active vibration control and robotics. This type of control concept originated from process control problems, which were summarized in Porter and Crossley [1]. Gould and Muarray-Lasso intended the modal control concepts to structure systems [2]. Meirovitch [3] developed an independent modal space control (IMSC) method for controlling a single mode of a distributed mass body. In IMSC, the equations of motion for the structure are decoupled using modal analysis and then modal control forces are determined by minimizing a performance index. Baz and Poh [4] in their study performed a numerical study on controlling the vibrations of a beam instrumented with piezoelectric patches, using IMSC. They modified the IMSC to control multiple modes and called the result modified independent modal space control (MIMSC). Baz and Poh [5] experimentally verified the theoretical developments by controlling the first two modes of a cantilevered beam. Other notable works can be found in Anthonis [6]. He applied modal space decoupling control to control the active suspension of a spray boom. The two hydraulic actuators of the active suspension are placed symmetrically on the spray boom. Because the spray boom itself is also symmetrical, the two excitations can be transformed to a translational and rotational excitation. This modal transformation yields perfect decoupling in the frequency range where only the rigid body modes determine the vibration. Lauwerys, Swevers and Sas [6] used a similar approach to control an active suspension of a car. The motion of the car body is transformed in heave, roll and pitch motions which are decoupled. Lin and Yu [7] used modal decoupling to suppress the vibration of a rotor. The popularity of modal space control is due to its numerous benefits. One of the most important advantages is its ability to reduce the order of a system by choosing the most important modes of the system. Also, the natural modes decouple the structural system and further simplify the system analysis, no optimization is required and numerically stable methods exist to determine the modal vectors. In addition, a tremendous insight is gained into the characteristics and behavior of the system [8]. While significant progress has been made in the field of modal space control concept, the application of modal space to the control of hydraulically driven 6 DOF motion simulation platform has not been fully exploited. In this study the application of modal space control strategy, which is receiving intense interest in the control of multi- degree-of-freedom (MDOF) systems and has been successively applied to a wide variety of practical problems, is examined. It is the purpose of this work to extend the modal space control to 6 DOF motion simulation platform by using modal space decoupling control strategy (MSDC). The methodology is based on exploitation of the properties of the joint space inverse
  • 2. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 116 This work is licensed under Creative Commons Attribution 4.0 International License. mass matrix. Through a mapping of the control and feedback variables, from the joint space to the decoupling modal space, the highly coupled MDOF dynamics is transformed into six independent single-input single-output (SISO) 1 DOF hydraulically driven mechanical system. II. SIX DEGREES OF FREEDOM PARALLEL MANIPULATOR (PM) DYNAMICS The 6 DOF PM is a parallel mechanism that consists of a rigid body top plate, or mobile plate, connected to a fixed base plate and is defined by at least three stationary points on the grounded base connected to six independent kinematic legs. The six legs of the 6 DOF PM are connected to both the base plate and the top plate by universal joints in parallel located at both ends of each leg. The legs are designed with an upper body and lower body that can be adjusted, allowing each leg to be varied in length. Each leg subsystem contains two bodies connected together with a cylindrical joint. The position and orientation of the mobile platform varies depending on the lengths to which the six legs are adjusted. The overall system has six degrees of freedom. The mechanism is depicted pictorially in figure 1. Lower gimbal point Moving platform (p) Servovalve Fixed base (b) Actuator Hydraulic cylinder Upper gimbal point Rod Figure 1: Hydraulically Driven 6 DOF Parallel Manipulator III. SIX DEGREES OF FREEDOM PM KINEMATICS The kinematics analysis that establishes the relationship between the lengths of the six actuators and the position of the moving platform will be presented. There are two main axis systems used to describe the motion: body-fixed and earth-fixed coordinates. The calculations are done in the body-fixed axis to simplify the equations; the results are then transformed to the earth- fixed coordinate system for proper updating of position and orientation. The earth-fixed system is considered an inertial reference frame, neglecting the rotational velocity of the earth and conforming to Newton’s laws of motion. A flat earth assumption is made, which essentially aligns the acceleration of gravity along the vertical earth-fixed system. In figure 3 a schematic drawing of the coordinate system is shown, addressing the geometric relations of the system with coordinates of 6 DOF PM. The six joints are on the center Ai (i = 1, 2... 6), located in the circle with radius ra, and denoted by A1, A3, A5 and A2, A4, A6. The center of the six joints is connected to form a hexagon. Similarly, six of the attachment points Bi (i = 1, 2..., 6), located in the circle of radius rb, and B1, B3, B2 and B4, B5, B6, respectively constitute two equilateral triangles. The connection of the upper and lower attachment points form a platform rotated 180 degrees with respect to each other. The motion of the moving platform is generated by actuating the cylindrical joints which vary the lengths of the legs, li, i = 1….6.
  • 3. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 117 This work is licensed under Creative Commons Attribution 4.0 International License. 6 A 1 B 5 A 4 A 3 A 2 l a d p O 2 A 2 B 3 B 4 B 5 B 6 B b d b r a r ) ( b p X X ) ( b p Y Y p Y p X b X b Y b Z p Z 1 A 2 A 3 B 2 B 4 B 5 B 1 B 6 B b O 2 l 1 2 3 4 5 6 1 2 3 4 5 6 p O 3 A 4 A 5 A 6 A 1 A 1 α 2 α a) Front view b) top view Figure 3: Schematic diagram of coordinate system of 6-DOF PM So, trajectory of the center point of moving platform is adjusted by using these variables. For modeling of the mechanism, a base reference frame {B} (OB-XB-YB- ZB) is defined as shown in figure 3. A second frame {p} (Op-Xp-Yp-Zp) is attached to the center of the moving platform, Op and the points linking the legs to the moving platform are noted as Ai, i = 1….6, and each leg is attached to the base platform at the point Bi, i = 1….6. At neutral pose the body axes {p} attached to the moving platform are parallel to and coincide with the inertial frame {B} fixed to the base with its origin at the geometric centre of the base platform. Kinematics and dynamics analysis of the 6 DOF parallel mechanisms has been well established and can be found in several literatures [9-13]. 3.1 Control Design The structure of the proposed modal space controller is similar to the conventional joint space control as shown in figure 4. However, the difference is that the signals including control errors, control outputs and pressure difference feedbacks are transformed into decoupled modal space, so that the coupling effects are fully considered and compensated. Thus, the proportional gains and dynamic pressure feedback functions can be tuned independently in modal space. Also, the bandwidth can be raised near to the undamped eigenfrequencies. Figure 4: Structure of the modal space decoupled controller   T des z y x x    , , , , ,  i p i i b t Ra l    Inverse Kinematics Servo Valve p K L P s 1 dp K l c  1 + - + - + - T U T U U Dynamic pressure feedback correction Actuator position Pressure feedback Desired Input reference Transformation matrix G J A K c T lq 2 1 q tc 1  + + Gravity force compensation
  • 4. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 118 This work is licensed under Creative Commons Attribution 4.0 International License. Based on this model, the control inputs of the servo valves can be developed. It can be expressed as: T T a,1 a,2 a,6 T ,1 ,2 ,6 T ,1 ,2 ,6 ,1 ,2 ,6 T 1 2 1 diag( ) s s s diag s 1 s 1 s 1 i c c c dp dp dp c c c tc lq q k k k k k k c K A                                  L i U U e U U P J G (1) com   e l l (2) Where e —— position error (m), com l i —— actuator length command in terms of inverse kinematics (m). And each element of i i can be give as 6 6 6 ,k L 2 1 1 1 , 1 s s 1 i c tc ik jk a,k i ik jk dp,k ij i i k k c k q c U U k U U k K A                                   i i e P J G (3) Where e —— i e (m), i U ——  , jk U L P —— Li P (Pa), 1 T - lq J ——  , ij J G —— ( ) i G (m/s2 ). When 6 , 2 , 1 , ... a a a k k k   and, 6 , 2 , 1 , ... dp dp dp k k k   and unitary orthogonal matrix, U equals one, then, the modal space controller degenerates into the conventional joint space controller. IV. RESULTS AND DISCUSSION In this section, a particular test case representing typical physical conditions are designed to demonstrate the performance and feasibility of the decoupling control strategy by integrating and testing our methodologies, and numerical simulation results from the PM is realized in the Matlab/Simulink environment. A comparison was being made between the modal space controller and the conventional controller to discuss the effectiveness or otherwise of both controllers. The evaluation of control performance of the 6 DOF PM as applicable to flight motion simulators, most often deals with the describing function. The describing function can be in the form of measuring the frequency response such that the magnitude of the response is flat. These description functions enable the characterization of some of the parameters which are considered most important in control such as bandwidth, damping and interaction effects in multivariable system [14]. The results of Bode diagrams of the closed loop system from the desired to actual positions applying the modal space controller are shown in figures 5 and 6 respectively.
  • 5. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 119 This work is licensed under Creative Commons Attribution 4.0 International License. Figure 5: Simulation result of modal space controlled motion system frequency response Figure 6: Simulation result of modal space controlled motion system frequency response There are several characteristics of the system that can be read directly from this Bode plot. For example, in figures 5 and 6, the -3dB point can be found around 71.2Hz for pitch (Ry) and roll (Rx), 66.7Hz for heave (z), 29.4Hz for yaw. In surge(x) and sway (y) the bandwidth is just only 18.1Hz which is the lowest compared to other directions. The -90 deg bandwidth can be found at 28.7Hz for pitch (Ry) and roll (Rx), 27.7Hz for heave (z), 17.2Hz for yaw (Rz), 11.2Hz for surge(x) and sway(y). Highest bandwidths are attained with pitch (Ry), roll (Rx) and lowest in surge(x) and sway(y). All responses are reasonably flat without peaking above +3dB. As mentioned earlier, our analysis and comparing is based on the performance of the controllers on hydraulically driven 10 1 10 2 -180 -90 0 Phase (deg) Frequency (Hz) -40 -30 -20 -10 0 10 Magnitude (dB) Ψ z φ,θ z x,y φ,θ x,y Ψ 10 1 10 2 -180 -90 0 Phase (deg) Frequency (Hz) -40 -30 -20 -10 0 10 Magnitude (dB) Ψ z φ,θ z x,y φ,θ x,y Ψ 10 1 10 2 -180 -90 0 Phase (deg) Frequency (Hz) -40 -30 Ma z φ,θ x,y Ψ 10 1 10 2 -180 -90 0 Phase (deg) Frequency (Hz) -40 -30 Ma z φ,θ x,y Ψ
  • 6. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 120 This work is licensed under Creative Commons Attribution 4.0 International License. 6 DOF PM as it relates to flight motion simulator. The results obtained from modal space control strategy are compared with conventional joint space controller for the dynamic responses of the closed loop system. The frequency response for the conventional joint space controller is given in Figures 7 and 8. Figure 7: Simulation result of conventional PID controlled motion system frequency response Figure 8: Simulation result of conventional PID controlled motion system frequency response Because there is no compensation of coupling effects for the different natural frequencies, the responses of surge (x) and sway (y), demonstrate peaks much more than others (such as pitch (Ry) and roll (Rx), which are over damped). Except for surge and pitch, the bandwidth is as low as 8.4Hz. Moreover, a compromise has to be found between peaking (up till 3dB) of the lowest bandwidth loops of surge (x) and sway (y) and the over damped response of pitch (Ry) and roll (Rx). For this control structure, the control action is being limited by the lowest eigenfrequencies of the system, therefore, the maximum attainable bandwidth is 8.5Hz for both roll (Rx) and pitch (Ry). With the conventional controller the attainable bandwidth for x and y is 21.8Hz, which is higher than with modal space controller which is only 18.1Hz. -40 -30 -20 -10 0 10 20 Magnitude (dB) 10 1 10 2 -270 -180 -90 0 Phase (deg) Frequency (Hz) x,y z φ,θ Ψ x,y Ψ z φ,θ -40 -30 -20 -10 0 10 20 Magnitude (dB) 10 1 10 2 -270 -180 -90 0 Phase (deg) Frequency (Hz) x,y z φ,θ Ψ x,y Ψ z φ,θ -40 -30 -20 -10 0 Magnitude (dB) 10 1 10 2 -270 -180 -90 0 Phase (deg) Frequency (Hz) x,y φ,θ Ψ x,y Ψ z φ,θ
  • 7. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 121 This work is licensed under Creative Commons Attribution 4.0 International License. Nevertheless, this does not suggest that the dynamic performance along these degrees of freedom is deteriorated using modal space controller. As can be observed in figure 9, the conventional control bandwidth is achieved at the expense of higher overshoot and poor performances of other degrees of freedom, however, a high percentage of overshot is an undesirable phenomenon in flight motion simulator. Moreover, the bandwidth of the modal space controller can be tuned to a higher level, as the same as the conventional control, but that will not meet the requirements of a flight simulator motion system in which flat frequency response is preferable [15]. In addition, with the modal space controller, obviously each degree of freedom can be almost tuned independently, and it is possible to raise their bandwidths near to the undamped eigenfrequencies. Comparing figure 5 and figure 6 we can see that the bandwidth of roll and pitch increased from 8.5Hz to 71.1Hz, yaw is raised from 8.4Hz to 29.4Hz, and heave increased from 9.2Hz to 66.7Hz. The comparison of two control strategies indicates that the modal space control performed better than the conventional. Transients’ response results as shown in figure 9 and 10 are also provided to show characteristics of the system in time domain. Step inputs are fed into the system along each DOF in proper sequence. Inputs amplitude is 1 deg along rotation axes and 3 mm along translation axes respectively. a) roll d) roll b) pitch e) pitch 1 1.5 2 2.5 -0.02 -0.01 0 0.01 0.02 Time (s) Amplitude (rad) 1 1.5 2 2.5 -0.02 -0.01 0 0.01 0.02 Time (s) Amplitude (rad) Ref    1 1.5 2 2.5 -0.02 -0.01 0 0.01 0.02 Time (s) Amplitude (rad) Ref    1 1.5 2 2.5 -0.02 -0.01 0 0.01 0.02 Time (s) Amplitude (rad)
  • 8. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 122 This work is licensed under Creative Commons Attribution 4.0 International License. c) yaw f) yaw Figure 9: Simulation result of short time response with step position inputs of the modal space control (left column, a-c) and conventional control (right column, d-f) a) surge d) surge b) sway e) sway 1 1.5 2 2.5 -0.02 -0.01 0 0.01 0.02 Time (s) Amplitude (rad) 1 1.5 2 2.5 -0.02 -0.01 0 0.01 0.02 Time (s) Amplitude (rad) 3 3.5 4 4.5 -4 -2 0 2 4 x 10 -3 Times (s) Amplitude (m) 1 1.5 2 2.5 -5 0 5 x 10 -3 Time (s) Amplitude (m) 3 3.5 4 4.5 -4 -2 0 2 4 x 10 -3 Times (s) Amplitude (m) 1 1.5 2 2.5 -5 0 5 x 10 -3 Time (s) Amplitude (m)
  • 9. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 123 This work is licensed under Creative Commons Attribution 4.0 International License. 3.5 4 5 -4 -2 0 4 Time (s) Amplitude (m) 2 x 10 -3 -3 4.5 c) heave f) heave Figure 10: Simulation result of short time response with step position inputs of the modal space control (left column, a-c) and conventional control (right column, d-f) V. CONCLUSION Modal space decoupled control for a hydraulically driven motion simulation platform has been presented. The approach is based on singular values decomposition to the properties of joint-space inverse mass matrix, and mapping of the control and feedback variables from the joint space to the decoupling modal space. The method transformed highly coupled six-input six-output dynamics into six independent single-input single-output (SISO) 1 DOF hydraulically driven mechanical systems. It has also been shown that the methods well suited for 1 DOF hydraulically driven mechanical systems can be applied to control a multi input multi output and highly coupled hydraulically driven 6 DOF motion simulation platform. Although, the structure of our proposed modal space controller is very similar to the conventional joint space control, the novelty in our approach is that the signals including control errors, control outputs and pressure feedbacks are transformed into decoupled modal space and also the proportional gains and dynamic pressure feedback are tuned in modal space. The results indicate that the conventional controller can only attenuate the resonance peaks of the lower eigenfrequencies of six rigid modes properly, and the peaking points of other relative higher eigenfrequencies are over damped, so the system bandwidth is limited by the lower eigenfrequencies. In addition, the results show that it is very effective to design and tune the system in modal space. The bandwidth increased substantially except surge (x) and sway (y) motions, each degree of freedom can be almost tuned independently and their bandwidths can be increased near to the undamped eigenfrequencies. REFERENCES [1] Porter, B. & Crossely, R. (1992). Modal control theory and application. London: Taylor and Francis. [2] Gould, L. A. & Murry-Lasso, M. A. (1966). On the modal control of distributed parameter systems with distributed feedback. IEEE Trans. Automatic Control, Ac- 11, 729-737. [3] Meirovitch, L. (1989). Dynamics and control of structures. New York: Wiley. [4] Baz A & Poh S. (1988). Performance of an active control system with piezoelectric actuators. J. Sound Vibration, 126, 327-343. [5] Baz, A. & Poh, S. (1999). Experimental implementation of the modified independent modal space control method. J. Sound Vibration, 139, 133-149. [6] Anthonis, J. & Ramon, H. (2003). Design of an active suspension to suppress the horizontal vibrations of a spray boom. Journal of Sound and Vibration, 266(3), 573-583. [7] Lauwerys, C., Swevers, J., & Sas, P. (2005). A model free control design approach for a semi-active suspension of a passenger car. In: Proceedings of the American Control Conference, Portland, USA, pp. 2206-2211. [8] Lin, Y.H. & Yu, H.-C. (2004). Active modal control of flexible rotor. Mechanical Systems and Signal Processing, 8(5), 1117-1131. [9] Xinjian Niu, Chifu Yang, Bowen Tian, Xiang Li, & Junwei Han. (2019). Modal decoupled dynamics feed- forward active force control of spatial multi-dof parallel robotic manipulator. Available at: https://doi.org/10.1155/2019/1835308. [10] Guo, H., Liu, Y. G. Liu, & G.R. et al. (2008). Cascade control of a hydraulically driven 6-dof parallel mechanism manipulator based on a sliding mode. Control Engineering Practice, 16, 1055-1068. 1 1.5 2 2.5 -4 -2 0 2 4 x 10 -3 Time (s) Amplitude (m)
  • 10. International Journal of Engineering and Management Research e-ISSN: 2250-0758 | p-ISSN: 2394-6962 Volume-12, Issue-1, (February 2022) www.ijemr.net https://doi.org/10.31033/ijemr.12.1.15 124 This work is licensed under Creative Commons Attribution 4.0 International License. [11] P. Ogbobe, Y. Zhengmao, H, Jiang, C. Yang, & J. Han. (2011). Formulation and evaluation of coupling effect between DOF motions of hydraulically driven 6 DOF parallel manipulator. Iranian Journal of Science & Technology, Transaction of Mechanical Engineering, 35(2), 143-175. DOI: 10.22099/ijstm.2011.902. [12] Zhang, C. & Jiang, H. (2021). Rigid-flexible modal analysis of the hydraulic 6-dof parallel mechanism. Energies, 14, 1604. https://doi. org/10.3390/en14061604. [13] Gosselin CM. (1998). A new approach for the dynamic analysis of parallel manipulator. Multibody System Dynamics, 2, 317. [14] Meirovitch, L. (1986). Elements of vibration analysis. (2nd ed.). New York, USA: McGraw-Hill. [15] Koekebakker, S.H. (2001). Model based control of a flight simulator motion system. PhD Thesis, Delft University Technology.